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REVIEW 5 major objections 6 minor 114 references

Learnable quantum spectral filters for hybrid graph neural networks

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A parameterized QFT circuit whose gate connections are read from the adjacency matrix can approximate a graph Laplacian's eigenspace, compressing graph signals to log(N) qubits.

desk verdict A genuinely new contraction-and-initialization idea with a clean experimental protocol sits on top of an encoding that provably discards node-0 edges, so the central spectral-filter claim fails. read the letter →

arxiv 2507.05640 v2 pith:BQSVF6OV submitted 2025-07-08 quant-ph cs.LG

Ammar Daskin This is my paper
classification quant-phcs.LG
keywords quantumspectralfiltersgraphneuralnetworksFouriertransformLaplacianhybridquantum-classicalmachinelearningclassificationexponentialcompressionparameterizedcircuits
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a parameterized quantum Fourier transform circuit, whose controlled gates are placed according to a simple contraction of the graph's adjacency matrix, can approximate the eigenspace of the graph Laplacian. If that is true, acting with the circuit on a flattened graph signal and measuring the qubits yields an $n$-dimensional probability vector that is at once a spectral filter and a pooling layer, compressing an $N$-node graph to about $\log(N)$ qubits. The author argues this removes the need for classical Chebyshev or Taylor polynomial approximations of spectral filters. The paper reports graph-classification accuracies on several TUDataset benchmarks that are comparable to, and sometimes better than, classical GNN baselines even though the quantum stage uses only 1--100 learnable parameters.

What carries the argument

The load-bearing object is the qubit connection matrix $M[c,t] = \sum_{i,j: (i)_2[c]=1,\, (j)_2[t]=1} A[i,j]/N$ (Eq.~7), which contracts an $N$-node adjacency matrix down to an $n \times n$ matrix on $\log(N)$ qubits. This matrix decides which controlled-$R_y$ and controlled-$R_z$ gates appear in each layer of the parameterized QFT and seeds their initial phases via $\phi_{c,t} = (1-\alpha_{\mathrm{init}})\,\mathrm{rand}[c,t] + \alpha_{\mathrm{init}} M[c,t]$. The circuit is trained to minimize the squared Frobenius norm of the off-diagonal part of $U^{\dagger} L U$, which pushes $U$ toward the Laplacian eigenbasis.

What would settle it

Take two non-isomorphic graphs that collapse to the same qubit-connection matrix under Eq. (7) and train the model on both; if the outputs are nearly identical, the contraction has discarded the spectral differences the method claims to preserve. A simpler check: retrain with the connection matrix replaced by random values; if classification accuracy does not change, the adjacency-derived connections are not what carries the result.

Watch

Extended reading notes

Core claim

The central claim is that the eigenspace of a graph's normalized Laplacian can be approximated by a QFT-based variational circuit whose connectivity is derived from the adjacency matrix. For an $N \times N$ Laplacian with $N = 2^n$, the circuit uses $n$ qubits and $O(l n^2)$ gates for $l$ layers; training drives the circuit unitary $U$ so that $U^{\dagger} L U$ is nearly diagonal, and the measurement statistics of the circuit on the flattened graph signal give an $n$-dimensional probability vector. The paper treats this vector as the filtered, exponentially compressed graph representation and feeds it to a small classical network for classification.

Load-bearing premise

The contraction in Eq. (7) preserves enough of the Laplacian's spectral information that a QFT circuit seeded by it can still filter the graph's frequencies; the paper gives no bound, invariant, or ablation showing this.

Editorial extensions

If this is right

  • A graph with $N$ nodes and $d$-dimensional features is processed with only $n = \log(Nd)$ qubits, so both the quantum layer and the classical head shrink exponentially with the graph size.
  • The quantum layer performs convolution and pooling in one step: the measured probability vector is the compressed, filtered signal that a classical network then classifies.
  • Classical Chebyshev or Taylor approximations of spectral filters become unnecessary; the filter is learned directly in the circuit parameters.
  • On datasets where geometry matters (MUTAG, Letter-medium/low, AIDS), accuracy stays within a few points of classical GNNs despite the exponential compression.
  • Increasing the number of layers raises the circuit's ability to diagonalize the Laplacian, but one or a few layers already suffice for learning tasks, and more layers risk overfitting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Replacing Eq. (7) with a random or uniform connection rule would test whether the spectral information in the adjacency matrix is actually load-bearing; the paper does not run this ablation.
  • The Fourier-like-eigenbasis assumption suits structured graphs such as molecules, paths, and cycles; for random or expander graphs, a QFT-based circuit may need many more layers, and the exponential compression may break down.
  • Because the circuit is shallow and the output is a standard-basis probability vector, the whole model may be classically simulable; the likely practical benefit is parameter efficiency and compression rather than a quantum speedup.
  • The same circuit family could be extended to node-level tasks by reading local expectation values (single-qubit or pair correlations) instead of the global probability vector.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper proposes a hybrid quantum-classical graph neural network in which an n-qubit parameterized quantum Fourier transform circuit is used as a convolutional and pooling layer. The controlled gates in the circuit are intended to be derived from an n×n qubit connection matrix M, constructed from the graph adjacency matrix via Eq. (7). The authors claim that the circuit can approximate the eigenspace of the graph Laplacian, that its measurement outcome is a filtered and exponentially compressed graph signal, and that a small classical head then performs graph classification. Experiments are reported on several TUDatasets and compared with classical GNN baselines.

Significance. If the central claims were established, the proposal would be notable: it suggests an exponential reduction from N-dimensional graph data to n=log(N) qubits with O(l n^2) gates, bypassing classical Chebyshev or Taylor spectral approximations. The paper also ships publicly available code and makes a concrete numerical study of circuit expressivity. However, the central claim that Eq. (7) preserves enough spectral structure for the circuit to act as a spectral filter is neither formalized nor supported by the experiments. The contraction has a concrete structural failure, the expressivity experiments are circular relative to the claim, and the classification results do not provide the stated validation.

major comments (5)
  1. [§3.1, Eq. (7)] The qubit connection matrix M is not faithful under vertex relabeling and demonstrably discards spectral information. Node 0 has binary string 00...0, so no edge incident to node 0 contributes to any M[c,t]; for the 4-node graph with edge set {(0,1)}, M is the zero matrix, making the circuit identical to the empty graph's circuit. The isomorphic graph with edge {(2,3)} gives nonzero M entries, so Eq. (7) is not invariant under relabeling. Since M is the only graph-dependent input to the circuit (apart from random noise added later), this is a load-bearing flaw in the claim that the circuit encodes the Laplacian eigenspace. No fidelity bound, invariance property, or ablation isolating Eq. (7) is provided.
  2. [§3.2, Eqs. (9)-(10)] The expressivity experiment is circular with respect to the central claim. The loss function optimizes U_circuit to minimize the off-diagonal norm of U† L U on the same graphs whose eigenspace approximation is then claimed; this demonstrates that the parameterized circuit can fit the eigenspace after optimization, not that Eq. (7) supplies a graph-derived approximation or a useful initialization. No error bound, convergence guarantee, or sample complexity is given, so the statement in §1.2(ii) that the circuit 'can approximate the eigenspace of any normalized Laplacian with high accuracy' is unsupported.
  3. [§3.3] The model is not shown to be a spectral filter. A spectral graph filter is defined as g(L) = U g(Λ) U^T, with a filter function applied to the eigenvalues (Eqs. (2)-(4)). The proposed layer is a parameterized unitary U(θ) acting on an encoded input state followed by measurement; no equation relates the circuit output to the graph Fourier transform U^T x or to a diagonal filter g(Λ). Without this relation, the claims that the circuit acts as a convolution-plus-pooling spectral filter and that the output is a filtered graph signal are not established.
  4. [Table 3] The reported comparisons do not support the statement that results are 'comparable to and in some cases better than many baseline results'. Baselines are drawn from multiple sources with different training protocols, and on ENZYMES the model achieves 33.0% versus 68.1% for GCN; on PROTEINS it achieves 67.0% versus 73.7% for DiffPool. The paper reports no paired significance tests and, for many baseline entries, no standard deviations. Additionally, the preprocessing step of adding 'small random noise to the connection matrices to connect every qubit' (Sec. 3.3) means the circuit used in the classification experiments is not determined by Eq. (7), so those experiments cannot validate the contraction.
  5. [§3.1.1] The admitted limitations contradict the universal claim in the abstract and §1.2(ii). The text states that sparse graphs can lead to an 'uninformative qubit connection matrix' and that the optimization gets stuck for the 2-qubit case in Fig. 3, yet the paper claims that the eigenspace of a graph Laplacian can be approximated for any graph. The limitation is stated only qualitatively, with no characterization of the class of graphs for which Eq. (7) is informative or of the approximation error in the cases where it is not.
minor comments (6)
  1. [§1.2(i)] The complexity statement 'O(l n^2) quantum gates and O(l n^2) parameters' should distinguish gate count from parameter count; the number of parameters in the rotation gates and the number of controlled gates are not necessarily equal.
  2. [§3.3 and Fig. 4] The text says the circuit output is obtained by measuring each qubit in the standard basis to obtain probabilities, but Fig. 4 labels the output as ⟨Z0⟩, ⟨Z1⟩, ⟨Z2⟩, which are Pauli-Z expectation values, not probabilities. Please clarify which quantity is actually used.
  3. [§2.1] References [83] and [84] appear twice verbatim in the related-work paragraph; the duplicate should be removed.
  4. [Eq. (7)] The notation '(i)_2[c] = 1' is undefined; please define it as the c-th bit of the binary representation of i, counting from the least significant bit, and state the bit-ordering convention.
  5. [Fig. 3] The figure would be more informative with labeled axes and a quantitative description of the loss scale; the text says the final loss is 'very small' without reporting numerical values.
  6. [§1.1] The sentence 'The Laplacian operators of graphs have Fourier-like eigenbasis' is imprecise; this holds for specific graph families (e.g., circulant graphs or paths) and should be stated with the necessary conditions.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the Sec. 3.2 'eigenspace approximation' is the optimizer's own training loss; classification benchmarks are external and not circular.

  1. fitted input called prediction [Sec. 3.2, Eqs. (9)-(10), Fig. 3; claim reiterated in Sec. 1.2(ii)]
    "During the optimization, the unitary for the circuit U^{(k)}_{circuit} is generated for the given parameters at iteration k by using matrix multiplication: RÂΛ(k)=U^{(k)†}_{circuit} L U^{(k)}_{circuit}. When U_circuit is the true eigenspace, then Λ̂ is a diagonal matrix containing the eigenvalues of L. The results in the figures show the loss function, calculated using the Frobenius norm of the off-diagonal elements of Λ̂(k): Loss_k = Σ_{i≠j} |Λ̂^{(k)}_{ij}|^2."

    The paper's evidence that 'the eigenspace of the graph Laplacians' can be approximated is the final value of Loss_k, which is the exact quantity minimized during the variational optimization on the same graphs. Eq. (10) defines approximation error as the off-diagonal norm; optimizing U against that objective and then reporting that the norm is small makes the result a fitted quantity, not a prediction or independent derivation. No held-out Laplacian is used, and no control with random or fixed connection matrices is provided, so the specific claim that Eq. (7)'s adjacency-derived connections are responsible for the spectral approximation is untested. Any sufficiently expressive parameterized circuit would drive this training loss down.

full rationale

The classification experiments in Sec. 4 are compared against external TUDataset baselines and are not circular. The central circular element is the expressivity demonstration: Sec. 3.2 optimizes U to minimize the off-diagonal norm of U† L U for the same L whose eigenspace approximation is claimed, so the small final loss is the optimizer's objective rather than a free-standing prediction. The paper's self-citations [54], [95], and [101] are present but not load-bearing, since the Fourier-like-spectra premise is also supported by external references [52,53] and the main performance claims rest on external benchmarks. The Eq. (7) connection rule has a serious correctness limitation not fully acknowledged: because M[c,t] is accumulated only when both bit c of i and bit t of j are 1, no edge incident to node 0 ever contributes to M, so some graphs produce an empty connection matrix; Sec. 3.1.1 notes only the sparse-row/column case. That is a validity concern rather than a circularity, so it is weighed in the score only insofar as it weakens the claimed connection between adjacency and the fitted spectral approximation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central architecture rests on three unproven premises: Fourier-like Laplacian eigenspaces, the fidelity of the Eq. (7) contraction, and the sufficiency of amplitude-embedded measurements. The listed hyperparameters directly shape the graph-to-circuit mapping and are chosen by hand. No new physical entities are introduced.

free parameters (3)
  • alpha_init (phase-inclusion ratio) = 0.1 in experiments (0.5 earlier in Sec. 3.1)
    Controls how much the graph-derived connection matrix contributes to initial CRZ phases; chosen by hand to avoid overfitting, not derived from data.
  • Quantum circuit layers nlayers = 4 for most datasets, 1 for MUTAG
    Hand-chosen per dataset; the paper notes more layers increase expressivity but can cause overfitting.
  • Random noise added to connection matrices = unspecified
    Added to all connection matrices to prevent overfitting and to connect every qubit; the magnitude is not reported.
assumptions (4)
  • domain assumption The eigenvectors of Laplacians of geometrically structured graphs are Fourier-like, so a parameterized QFT circuit can approximate them with few layers.
    Invoked in Sec. 3 (paragraph 2) to justify the ansatz; not proven for the datasets used.
  • ad hoc to paper The connection matrix contraction in Eq. (7) preserves enough spectral structure for classification.
    This mapping from N-node adjacency to an n x n matrix is introduced for this paper and is not shown to be information-preserving.
  • domain assumption Amplitude-embedding of the flattened node features and measuring probabilities retains class-discriminative information.
    Implicit in the workflow of Sec. 3.3; no analysis of information loss from pooling to n probabilities is given.
  • standard math Spectral graph convolution duality x*g = U g(Lambda) U^T x (Eq. 3) is standard.
    Used as background in Sec. 2 to motivate spectral filters; not the source of the paper's new claims.

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Cite this review

Pith. "Pith review of Learnable quantum spectral filters for hybrid graph neural networks." pith.science (2026). https://pith.science/paper/BQSVF6OV

@misc{pith2026250705640,
  author       = {Pith},
  title        = {Pith review of: Learnable quantum spectral filters for hybrid graph neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQSVF6OV}},
  note         = {Machine review of arXiv:2507.05640}
}
abstract

In this paper, we describe a parameterized quantum circuit that can be considered as convolutional and pooling layers for graph neural networks. The circuit incorporates the parameterized quantum Fourier circuit where the qubit connections for the controlled gates derived from the Laplacian operator. Specifically, we show that the eigenspace of the Laplacian operator of a graph can be approximated by using QFT based circuit whose connections are determined from the adjacency matrix. For an $N\times N$ Laplacian, this approach yields an approximate polynomial-depth circuit requiring only $n=log(N)$ qubits. These types of circuits can eliminate the expensive classical computations for approximating the learnable functions of the Laplacian through Chebyshev polynomial or Taylor expansions. Using this circuit as a convolutional layer provides an $n-$ dimensional probability vector that can be considered as the filtered and compressed graph signal. Therefore, the circuit along with the measurement can be considered a very efficient convolution plus pooling layer that transforms an $N$-dimensional signal input into $n-$dimensional signal with an exponential compression. We then apply a classical neural network prediction head to the output of the circuit to construct a complete graph neural network. Since the circuit incorporates geometric structure through its graph connection-based approach, we present graph classification results for the benchmark datasets listed in TUDataset library. Using only [1-100] learnable parameters for the quantum circuit and minimal classical layers (1000-5000 parameters) in a generic setting, the obtained results are comparable to and in some cases better than many of the baseline results, particularly for the cases when geometric structure plays a significant role.

Figures

Figures reproduced from arXiv: 2507.05640 by the authors.

Figure 1
Figure 1. General workflow of hybrid (quantum-classical) graph neural network. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. One layer of the four-qubit parameterized connection-based quantum Fourier circuit for approx [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The loss function for 10 randomly generated Erd˝os–R´enyi graphs with edge probability 0.3, using [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: An example workflow of a quantum graph neural network. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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Reference graph

Works this paper leans on

114 extracted references · 51 canonical work pages

  1. [1]

    Submission category by year, 2025

    arXiv. Submission category by year, 2025. Accessed: 17 June 2025

    2025

  2. [2]

    Gpt-4 technical report

    Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv preprint arXiv:2303.08774, 2023

    arXiv 2023

  3. [3]

    Deepseek-v3 technical report.arXiv preprint arXiv:2412.19437, 2024

    Aixin Liu, Bei Feng, Bing Xue, Bingxuan Wang, Bochao Wu, Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, et al. Deepseek-v3 technical report.arXiv preprint arXiv:2412.19437, 2024

    arXiv 2024

  4. [4]

    A comprehensive overview of large language models

    Humza Naveed, Asad Ullah Khan, Shi Qiu, Muhammad Saqib, Saeed Anwar, Muhammad Usman, Naveed Akhtar, Nick Barnes, and Ajmal Mian. A comprehensive overview of large language models. arXiv preprint arXiv:2307.06435, 2023

    arXiv 2023

  5. [5]

    Welcome to the era of chatgpt et al

    Timm Teubner, Christoph M Flath, Christof Weinhardt, Wil Van Der Aalst, and Oliver Hinz. Welcome to the era of chatgpt et al. the prospects of large language models.Business & Information Systems Engineering, 65(2):95–101, 2023

    2023

  6. [6]

    A survey of sustainability in large language models: Applications, economics, and challenges

    Aditi Singh, Nirmal Prakashbhai Patel, Abul Ehtesham, Saket Kumar, and Tala Talaei Khoei. A survey of sustainability in large language models: Applications, economics, and challenges. In2025 IEEE 15th Annual Computing and Communication Workshop and Conference (CCWC), pages 00008– 00014. IEEE, 2025

    2025

  7. [7]

    Randomized algorithms.ACM Computing Surveys (CSUR), 28(1):33–37, 1996

    Rajeev Motwani and Prabhakar Raghavan. Randomized algorithms.ACM Computing Surveys (CSUR), 28(1):33–37, 1996

    1996

  8. [8]

    Randomized algorithms for matrices and data.Foundations and Trends® in Machine Learning, 3(2):123–224, 2011

    Michael W Mahoney et al. Randomized algorithms for matrices and data.Foundations and Trends® in Machine Learning, 3(2):123–224, 2011

    2011

Show all 114 references
  1. [9]

    A survey of randomized algorithms for training neural networks.Information Sciences, 364:146–155, 2016

    Le Zhang and Ponnuthurai N Suganthan. A survey of randomized algorithms for training neural networks.Information Sciences, 364:146–155, 2016

    2016

  2. [10]

    Springer, 2001

    Vijay V Vazirani.Approximation algorithms, volume 1. Springer, 2001

    2001

  3. [11]

    Cambridge university press, 2011

    David P Williamson and David B Shmoys.The design of approximation algorithms. Cambridge university press, 2011

    2011

  4. [12]

    Which problems have strongly exponential complexity?Journal of Computer and System Sciences, 63(4):512–530, 2001

    Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity?Journal of Computer and System Sciences, 63(4):512–530, 2001

    2001

  5. [13]

    Computer architecture and amdahl’s law.Computer, 46(12):38–46, 2013

    Gene M Amdahl. Computer architecture and amdahl’s law.Computer, 46(12):38–46, 2013

    2013

  6. [14]

    Amdahl’s law in the multicore era.Computer, 41(7):33–38, 2008

    Mark D Hill and Michael R Marty. Amdahl’s law in the multicore era.Computer, 41(7):33–38, 2008

    2008

  7. [15]

    Qwen3 technical report.arXiv preprint arXiv:2505.09388, 2025

    An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, et al. Qwen3 technical report.arXiv preprint arXiv:2505.09388, 2025

    2025 arXiv

  8. [16]

    Num- ber 47

    Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi.Classical and quantum computation. Num- ber 47. American Mathematical Soc., 2002

    2002

  9. [17]

    Expressive power of parametrized quantum circuits.Physical Review Research, 2(3):033125, 2020

    Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao. Expressive power of parametrized quantum circuits.Physical Review Research, 2(3):033125, 2020

    2020

  10. [18]

    Expressivity of quantum neural networks.Physical Review Research, 3(3):L032049, 2021

    Yadong Wu, Juan Yao, Pengfei Zhang, and Hui Zhai. Expressivity of quantum neural networks.Physical Review Research, 3(3):L032049, 2021

    2021

  11. [19]

    The power of quantum neural networks.Nature Computational Science, 1(6):403–409, 2021

    Amira Abbas, David Sutter, Christa Zoufal, Aur´ elien Lucchi, Alessio Figalli, and Stefan Woerner. The power of quantum neural networks.Nature Computational Science, 1(6):403–409, 2021

    2021

  12. [20]

    High-expressibility quantum neural networks using only classical resources

    Marco Maronese, Francesco Ferrari, Matteo Vandelli, and Daniele Dragoni. High-expressibility quantum neural networks using only classical resources. 2025. 13

    2025

  13. [21]

    A new model for learning in graph domains

    Marco Gori, Gabriele Monfardini, and Franco Scarselli. A new model for learning in graph domains. InProceedings. 2005 IEEE international joint conference on neural networks, 2005., volume 2, pages 729–734. IEEE, 2005

    2005

  14. [22]

    Graph neural networks for ranking web pages

    Franco Scarselli, Sweah Liang Yong, Marco Gori, Markus Hagenbuchner, Ah Chung Tsoi, and Marco Maggini. Graph neural networks for ranking web pages. InThe 2005 IEEE/WIC/ACM International Conference on Web Intelligence (WI’05), pages 666–672. IEEE, 2005

    2005

  15. [23]

    The graph neural network model.IEEE Transactions on Neural Networks, 20(1):61–80, 2009

    Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model.IEEE Transactions on Neural Networks, 20(1):61–80, 2009

    2009

  16. [24]

    Learning skillful medium- range global weather forecasting.Science, 382(6677):1416–1421, 2023

    Remi Lam, Alvaro Sanchez-Gonzalez, Matthew Willson, Peter Wirnsberger, Meire Fortunato, Ferran Alet, Suman Ravuri, Timo Ewalds, Zach Eaton-Rosen, Weihua Hu, et al. Learning skillful medium- range global weather forecasting.Science, 382(6677):1416–1421, 2023

    2023

  17. [25]

    A gentle intro- duction to graph neural networks.Distill, 6(9):e33, 2021

    Benjamin Sanchez-Lengeling, Emily Reif, Adam Pearce, and Alexander B Wiltschko. A gentle intro- duction to graph neural networks.Distill, 6(9):e33, 2021

    2021

  18. [26]

    Graph neural networks: A review of methods and applications.AI open, 1:57–81, 2020

    Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph neural networks: A review of methods and applications.AI open, 1:57–81, 2020

    2020

  19. [27]

    Graph neural networks: Taxonomy, advances, and trends.ACM Transactions on Intelligent Systems and Technology (TIST), 13(1):1–54, 2022

    Yu Zhou, Haixia Zheng, Xin Huang, Shufeng Hao, Dengao Li, and Jumin Zhao. Graph neural networks: Taxonomy, advances, and trends.ACM Transactions on Intelligent Systems and Technology (TIST), 13(1):1–54, 2022

    2022

  20. [28]

    Semi-supervised classification with graph convolutional networks

    Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016

    2016 arXiv

  21. [29]

    Inductive representation learning on large graphs

    Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. Advances in neural information processing systems, 30, 2017

    2017

  22. [30]

    Graph attention networks.stat, 1050(20):10–48550, 2017

    Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, Yoshua Bengio, et al. Graph attention networks.stat, 1050(20):10–48550, 2017

    2017

  23. [31]

    Gradient-based learning applied to document recognition.Proceedings of the IEEE, 86(11):2278–2324, 2002

    Yann LeCun, L´ eon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition.Proceedings of the IEEE, 86(11):2278–2324, 2002

    2002

  24. [32]

    Deep learning.nature, 521(7553):436–444, 2015

    Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning.nature, 521(7553):436–444, 2015

    2015

  25. [33]

    Convolutional neural networks on graphs with fast localized spectral filtering.Advances in neural information processing systems, 29, 2016

    Micha¨ el Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering.Advances in neural information processing systems, 29, 2016

    2016

  26. [34]

    Spectral networks and locally con- nected networks on graphs.arXiv preprint arXiv:1312.6203, 2013

    Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally con- nected networks on graphs.arXiv preprint arXiv:1312.6203, 2013

    2013 arXiv

  27. [35]

    Understanding convolutions on graphs.Distill, 6(9):e32, 2021

    Ameya Daigavane, Balaraman Ravindran, and Gaurav Aggarwal. Understanding convolutions on graphs.Distill, 6(9):e32, 2021

    2021

  28. [36]

    Simplifying graph convolutional networks

    Felix Wu, Amauri Souza, Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Weinberger. Simplifying graph convolutional networks. InInternational conference on machine learning, pages 6861–6871. Pmlr, 2019

    2019

  29. [37]

    Graph signal processing for machine learning: A review and new perspectives.IEEE Signal processing magazine, 37(6):117–127, 2020

    Xiaowen Dong, Dorina Thanou, Laura Toni, Michael Bronstein, and Pascal Frossard. Graph signal processing for machine learning: A review and new perspectives.IEEE Signal processing magazine, 37(6):117–127, 2020

    2020

  30. [38]

    Understanding spectral graph neural network.arXiv preprint arXiv:2012.06660, 2020

    Xinye Chen. Understanding spectral graph neural network.arXiv preprint arXiv:2012.06660, 2020

    2012 arXiv

  31. [39]

    Graphs, convolutions, and neural networks: From graph filters to graph neural networks.IEEE Signal Processing Magazine, 37(6):128– 138, 2020

    Fernando Gama, Elvin Isufi, Geert Leus, and Alejandro Ribeiro. Graphs, convolutions, and neural networks: From graph filters to graph neural networks.IEEE Signal Processing Magazine, 37(6):128– 138, 2020. 14

    2020

  32. [40]

    A survey on spectral graph neural networks.arXiv preprint arXiv:2302.05631, 2023

    Deyu Bo, Xiao Wang, Yang Liu, Yuan Fang, Yawen Li, and Chuan Shi. A survey on spectral graph neural networks.arXiv preprint arXiv:2302.05631, 2023

    2023 arXiv

  33. [41]

    Taylornet: A novel approach for spectral filter learning on graph data.Neurocomputing, 605:128358, 2024

    Liwen Xu, Jiali Chen, Zhonghua Han, and Yongxia Zhang. Taylornet: A novel approach for spectral filter learning on graph data.Neurocomputing, 605:128358, 2024

    2024

  34. [42]

    Universal quantum circuit for n-qubit quantum gate: A programmable quantum gate.arXiv preprint quant-ph/0602174, 2006

    Paulo BM Sousa and Rubens Viana Ramos. Universal quantum circuit for n-qubit quantum gate: A programmable quantum gate.arXiv preprint quant-ph/0602174, 2006

    2006 arXiv

  35. [43]

    Universal programmable quantum circuit schemes to emulate an operator.The Journal of chemical physics, 137(23), 2012

    Anmer Daskin, Ananth Grama, Giorgos Kollias, and Sabre Kais. Universal programmable quantum circuit schemes to emulate an operator.The Journal of chemical physics, 137(23), 2012

    2012

  36. [44]

    A universal quantum circuit scheme for finding complex eigenvalues.Quantum information processing, 13:333–353, 2014

    Anmer Daskin, Ananth Grama, and Sabre Kais. A universal quantum circuit scheme for finding complex eigenvalues.Quantum information processing, 13:333–353, 2014

    2014

  37. [45]

    Parameterized quantum circuits as machine learning models.Quantum science and technology, 4(4):043001, 2019

    Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. Parameterized quantum circuits as machine learning models.Quantum science and technology, 4(4):043001, 2019

    2019

  38. [46]

    A variational eigenvalue solver on a photonic quantum processor

    Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Al´ an Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5(1):4213, 2014

    2014

  39. [47]

    The theory of varia- tional hybrid quantum-classical algorithms.New Journal of Physics, 18(2):023023, 2016

    Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Al´ an Aspuru-Guzik. The theory of varia- tional hybrid quantum-classical algorithms.New Journal of Physics, 18(2):023023, 2016

    2016

  40. [48]

    Quantum algorithms for fixed qubit architectures.arXiv preprint arXiv:1703.06199, 2017

    Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Hartmut Neven. Quantum algorithms for fixed qubit architectures.arXiv preprint arXiv:1703.06199, 2017

    2017 arXiv

  41. [49]

    Natural parametrized quantum circuit.Physical Review A, 106(5):052611, 2022

    Tobias Haug and MS Kim. Natural parametrized quantum circuit.Physical Review A, 106(5):052611, 2022

    2022

  42. [50]

    From graphs to qubits: A critical review of quantum graph neural networks.arXiv preprint arXiv:2408.06524, 2024

    Andrea Ceschini, Francesco Mauro, Francesca De Falco, Alessandro Sebastianelli, Alessio Verdone, Antonello Rosato, Bertrand Le Saux, Massimo Panella, Paolo Gamba, and Silvia L Ullo. From graphs to qubits: A critical review of quantum graph neural networks.arXiv preprint arXiv:...

    2024 arXiv

  43. [51]

    Quantum graph learning: Frontiers and outlook.arXiv preprint arXiv:2302.00892, 2023

    Shuo Yu, Ciyuan Peng, Yingbo Wang, Ahsan Shehzad, Feng Xia, and Edwin R Hancock. Quantum graph learning: Frontiers and outlook.arXiv preprint arXiv:2302.00892, 2023

    2023 arXiv

  44. [52]

    How can we naturally order and organize graph laplacian eigenvectors? In2018 IEEE Statistical Signal Processing Workshop (SSP), pages 483–487

    Naoki Saito. How can we naturally order and organize graph laplacian eigenvectors? In2018 IEEE Statistical Signal Processing Workshop (SSP), pages 483–487. IEEE, 2018

    2018

  45. [53]

    The discrete cosine transform.SIAM review, 41(1):135–147, 1999

    Gilbert Strang. The discrete cosine transform.SIAM review, 41(1):135–147, 1999

    1999

  46. [54]

    Quantum simulations based on parameterized circuit of an antisymmetric matrix

    Ammar Daskin. Quantum simulations based on parameterized circuit of an antisymmetric matrix. arXiv preprint arXiv:2505.01023, 2025

    2025 arXiv

  47. [55]

    American Mathematical Soc., 1997

    Fan RK Chung.Spectral graph theory, volume 92. American Mathematical Soc., 1997

    1997

  48. [56]

    Laplacian matrices of graphs: a survey.Linear algebra and its applications, 197:143–176, 1994

    Russell Merris. Laplacian matrices of graphs: a survey.Linear algebra and its applications, 197:143–176, 1994

    1994

  49. [57]

    Algorithms, graph theory, and linear equations in laplacian matrices

    Daniel A Spielman. Algorithms, graph theory, and linear equations in laplacian matrices. InProceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, pages 2698–2722. World S...

    2010

  50. [58]

    A tutorial on spectral clustering.Statistics and computing, 17:395–416, 2007

    Ulrike Von Luxburg. A tutorial on spectral clustering.Statistics and computing, 17:395–416, 2007

    2007

  51. [59]

    Quantum spectral clustering through a biased phase estimation algorithm.TWMS Journal of Applied and Engineering Mathematics, 10(1):24–33, 2017

    Ammar Daskin. Quantum spectral clustering through a biased phase estimation algorithm.TWMS Journal of Applied and Engineering Mathematics, 10(1):24–33, 2017

    2017

  52. [60]

    Convergence of laplacian eigenmaps.Advances in neural information processing systems, 19, 2006

    Mikhail Belkin and Partha Niyogi. Convergence of laplacian eigenmaps.Advances in neural information processing systems, 19, 2006

    2006

  53. [61]

    Manifold learning: What, how, and why.Annual Review of Statistics and Its Application, 11(1):393–417, 2024

    Marina Meil˘ a and Hanyu Zhang. Manifold learning: What, how, and why.Annual Review of Statistics and Its Application, 11(1):393–417, 2024. 15

    2024

  54. [62]

    A user guide to low-pass graph signal pro- cessing and its applications: Tools and applications.IEEE Signal Processing Magazine, 37(6):74–85, 2020

    Raksha Ramakrishna, Hoi-To Wai, and Anna Scaglione. A user guide to low-pass graph signal pro- cessing and its applications: Tools and applications.IEEE Signal Processing Magazine, 37(6):74–85, 2020

    2020

  55. [63]

    An introduction to convolutional neural networks.arXiv preprint arXiv:1511.08458, 2015

    Keiron O’shea and Ryan Nash. An introduction to convolutional neural networks.arXiv preprint arXiv:1511.08458, 2015

    2015 arXiv

  56. [64]

    Introduction to convolutional neural networks.National Key Lab for Novel Software Technology

    Jianxin Wu. Introduction to convolutional neural networks.National Key Lab for Novel Software Technology. Nanjing University. China, 5(23):495, 2017

    2017

  57. [65]

    A survey of convolutional neural networks: analysis, applications, and prospects.IEEE transactions on neural networks and learning systems, 33(12):6999–7019, 2021

    Zewen Li, Fan Liu, Wenjie Yang, Shouheng Peng, and Jun Zhou. A survey of convolutional neural networks: analysis, applications, and prospects.IEEE transactions on neural networks and learning systems, 33(12):6999–7019, 2021

    2021

  58. [66]

    Review of lightweight deep convolutional neural networks.Archives of Computational Methods in Engineering, 31(4):1915–1937, 2024

    Fanghui Chen, Shouliang Li, Jiale Han, Fengyuan Ren, and Zhen Yang. Review of lightweight deep convolutional neural networks.Archives of Computational Methods in Engineering, 31(4):1915–1937, 2024

    1915

  59. [67]

    Convolutional neural networks on graphs with chebyshev approximation, revisited.Advances in neural information processing systems, 35:7264–7276, 2022

    Mingguo He, Zhewei Wei, and Ji-Rong Wen. Convolutional neural networks on graphs with chebyshev approximation, revisited.Advances in neural information processing systems, 35:7264–7276, 2022

    2022

  60. [68]

    Graph neural network, chebnet, graph convolutional network, and graph autoencoder: Tutorial and survey

    Benyamin Ghojogh and Ali Ghodsi. Graph neural network, chebnet, graph convolutional network, and graph autoencoder: Tutorial and survey. 2024

    2024

  61. [69]

    Spectral representations for convolutional neural networks.Advances in neural information processing systems, 28, 2015

    Oren Rippel, Jasper Snoek, and Ryan P Adams. Spectral representations for convolutional neural networks.Advances in neural information processing systems, 28, 2015

    2015

  62. [70]

    On the stability of polynomial spectral graph fil- ters

    Henry Kenlay, Dorina Thanou, and Xiaowen Dong. On the stability of polynomial spectral graph fil- ters. InICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5350–5354. IEEE, 2020

    2020

  63. [71]

    JHU press, 2013

    Gene H Golub and Charles F Van Loan.Matrix computations. JHU press, 2013

    2013

  64. [72]

    Wavelets on graphs via spectral graph theory.Applied and Computational Harmonic Analysis, 30(2):129–150, 2011

    David K Hammond, Pierre Vandergheynst, and R´ emi Gribonval. Wavelets on graphs via spectral graph theory.Applied and Computational Harmonic Analysis, 30(2):129–150, 2011

    2011

  65. [73]

    Revisiting convolutional neural network on graphs with polynomial approximations of laplace–beltrami spectral filtering.Neural Computing and Applications, 33:13693–13704, 2021

    Shih-Gu Huang, Moo K Chung, Anqi Qiu, and Alzheimer’s Disease Neuroimaging Initiative. Revisiting convolutional neural network on graphs with polynomial approximations of laplace–beltrami spectral filtering.Neural Computing and Applications, 33:13693–13704, 2021

    2021

  66. [74]

    Quantum convolutional neural networks.Nature Physics, 15(12):1273–1278, 2019

    Iris Cong, Soonwon Choi, and Mikhail D Lukin. Quantum convolutional neural networks.Nature Physics, 15(12):1273–1278, 2019

    2019

  67. [75]

    A tutorial on quantum convolutional neural networks (qcnn)

    Seunghyeok Oh, Jaeho Choi, and Joongheon Kim. A tutorial on quantum convolutional neural networks (qcnn). In2020 International Conference on Information and Communication Technology Convergence (ICTC), pages 236–239. IEEE, 2020

    2020

  68. [76]

    Quantum convo- lutional neural networks for high energy physics data analysis.Physical Review Research, 4(1):013231, 2022

    Samuel Yen-Chi Chen, Tzu-Chieh Wei, Chao Zhang, Haiwang Yu, and Shinjae Yoo. Quantum convo- lutional neural networks for high energy physics data analysis.Physical Review Research, 4(1):013231, 2022

    2022

  69. [77]

    Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases.Nature communications, 13(1):4144, 2022

    Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A McMahon, Colin Scarato, Fran¸ cois Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, et al. Realizing quantum convolutional neural networks on a superconducting quantum processo...

    2022

  70. [78]

    What can we learn from quantum convolutional neural networks?Advanced Quantum Technologies, page 2400325, 2023

    Chukwudubem Umeano, Annie E Paine, Vincent E Elfving, and Oleksandr Kyriienko. What can we learn from quantum convolutional neural networks?Advanced Quantum Technologies, page 2400325, 2023

    2023

  71. [79]

    Hybrid quantum-classical convolutional neural networks.Science China Physics, Mechanics & Astronomy, 64(9):290311, 2021

    Junhua Liu, Kwan Hui Lim, Kristin L Wood, Wei Huang, Chu Guo, and He-Liang Huang. Hybrid quantum-classical convolutional neural networks.Science China Physics, Mechanics & Astronomy, 64(9):290311, 2021. 16

    2021

  72. [80]

    Quantum convolutional neural networks for multi-channel supervised learning.Quantum Machine Intelligence, 5(2):41, 2023

    Anthony M Smaldone, Gregory W Kyro, and Victor S Batista. Quantum convolutional neural networks for multi-channel supervised learning.Quantum Machine Intelligence, 5(2):41, 2023

    2023

  73. [81]

    Classical-to-quantum convolutional neural network transfer learning.Neurocomputing, 555:126643, 2023

    Juhyeon Kim, Joonsuk Huh, and Daniel K Park. Classical-to-quantum convolutional neural network transfer learning.Neurocomputing, 555:126643, 2023

    2023

  74. [82]

    Quantum convolutional neural network based on variational quantum circuits.Optics Communications, 550:129993, 2024

    Li-Hua Gong, Jun-Jie Pei, Tian-Feng Zhang, and Nan-Run Zhou. Quantum convolutional neural network based on variational quantum circuits.Optics Communications, 550:129993, 2024

    2024

  75. [83]

    Quantum algorithms for deep convolutional neural networks.arXiv preprint arXiv:1911.01117, 2019

    Iordanis Kerenidis, Jonas Landman, and Anupam Prakash. Quantum algorithms for deep convolutional neural networks.arXiv preprint arXiv:1911.01117, 2019

    1911 arXiv

  76. [84]

    Quantum optical convolutional neural network: a novel image recognition framework for quantum computing.IEEE access, 9:103337–103346, 2021

    Rishab Parthasarathy and Rohan T Bhowmik. Quantum optical convolutional neural network: a novel image recognition framework for quantum computing.IEEE access, 9:103337–103346, 2021

    2021

  77. [85]

    Quantum convolutional neural networks are (effectively) classically simulable.arXiv preprint arXiv:2408.12739, 2024

    Pablo Bermejo, Paolo Braccia, Manuel S Rudolph, Zo¨ e Holmes, Lukasz Cincio, and M Cerezo. Quantum convolutional neural networks are (effectively) classically simulable.arXiv preprint arXiv:2408.12739, 2024

    2024 arXiv

  78. [86]

    Efficient classical simulation of random shallow 2d quantum circuits.Physical Review X, 12(2):021021, 2022

    John C Napp, Rolando L La Placa, Alexander M Dalzell, Fernando GSL Brandao, and Aram W Harrow. Efficient classical simulation of random shallow 2d quantum circuits.Physical Review X, 12(2):021021, 2022

    2022

  79. [87]

    Quantum graph neural networks.arXiv preprint arXiv:1909.12264, 2019

    Guillaume Verdon, Trevor McCourt, Enxhell Luzhnica, Vikash Singh, Stefan Leichenauer, and Jack Hidary. Quantum graph neural networks.arXiv preprint arXiv:1909.12264, 2019

    1909 arXiv

  80. [88]

    A quantum approximate optimization algorithm

    Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014

    2014 arXiv

  81. [89]

    From the quantum approximate optimization algorithm to a quantum alternating operator ansatz.Algorithms, 12(2):34, 2019

    Stuart Hadfield, Zhihui Wang, Bryan O’gorman, Eleanor G Rieffel, Davide Venturelli, and Rupak Biswas. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz.Algorithms, 12(2):34, 2019

    2019

  82. [90]

    Quantum-based subgraph convolutional neural networks.Pattern Recognition, 88:38–49, 2019

    Zhihong Zhang, Dongdong Chen, Jianjia Wang, Lu Bai, and Edwin R Hancock. Quantum-based subgraph convolutional neural networks.Pattern Recognition, 88:38–49, 2019

    2019

  83. [91]

    On the design of quantum graph convolutional neural network in the nisq-era and beyond

    Zhirui Hu, Jinyang Li, Zhenyu Pan, Shanglin Zhou, Lei Yang, Caiwen Ding, Omer Khan, Tong Geng, and Weiwen Jiang. On the design of quantum graph convolutional neural network in the nisq-era and beyond. In2022 IEEE 40th International Conference on Computer Design (ICCD), pages 2...

    2022

  84. [92]

    A quantum spatial graph convolu- tional neural network model on quantum circuits.IEEE Transactions on Neural Networks and Learning Systems, 2024

    Jin Zheng, Qing Gao, Maciej Ogorza lek, Jinhu L¨ u, and Yue Deng. A quantum spatial graph convolu- tional neural network model on quantum circuits.IEEE Transactions on Neural Networks and Learning Systems, 2024

    2024

  85. [93]

    Financial fraud detection using quantum graph neural networks.Quantum Machine Intelligence, 6(1):7, 2024

    Nouhaila Innan, Abhishek Sawaika, Ashim Dhor, Siddhant Dutta, Sairupa Thota, Husayn Gokal, Nandan Patel, Muhammad Al-Zafar Khan, Ioannis Theodonis, and Mohamed Bennai. Financial fraud detection using quantum graph neural networks.Quantum Machine Intelligence, 6(1):7, 2024

    2024

  86. [94]

    Quantum graph neural network models for materials search.Materials, 16(12):4300, 2023

    Ju-Young Ryu, Eyuel Elala, and June-Koo Kevin Rhee. Quantum graph neural network models for materials search.Materials, 16(12):4300, 2023

    2023

  87. [95]

    A unifying primary framework for qgnns from quantum graph states.The European Physical Journal Special Topics, pages 1–10, 2024

    Ammar Daskin. A unifying primary framework for qgnns from quantum graph states.The European Physical Journal Special Topics, pages 1–10, 2024

    2024

  88. [96]

    Ground state-based quantum feature maps.arXiv preprint arXiv:2404.07174, 2024

    Chukwudubem Umeano and Oleksandr Kyriienko. Ground state-based quantum feature maps.arXiv preprint arXiv:2404.07174, 2024

    2024 arXiv

  89. [97]

    Iterative quantum feature maps.arXiv preprint arXiv:2506.19461, 2025

    Nasa Matsumoto, Quoc Hoan Tran, Koki Chinzei, Yasuhiro Endo, and Hirotaka Oshima. Iterative quantum feature maps.arXiv preprint arXiv:2506.19461, 2025

    2025 arXiv

  90. [98]

    Efficient quantum feature extraction for cnn-based learning

    Tong Dou, Guofeng Zhang, and Wei Cui. Efficient quantum feature extraction for cnn-based learning. Journal of the Franklin Institute, 360(11):7438–7456, 2023. 17

    2023

  91. [99]

    Hybrid quantum- classical graph convolutional network.arXiv preprint arXiv:2101.06189, 2021

    Samuel Yen-Chi Chen, Tzu-Chieh Wei, Chao Zhang, Haiwang Yu, and Shinjae Yoo. Hybrid quantum- classical graph convolutional network.arXiv preprint arXiv:2101.06189, 2021

    2021 arXiv

  92. [100]

    Quantum graph as a quantum spectral filter.Journal of Mathematical Physics, 54(3), 2013

    Ondˇ rej Turek and Taksu Cheon. Quantum graph as a quantum spectral filter.Journal of Mathematical Physics, 54(3), 2013

    2013

  93. [101]

    Quantum rnns and lstms through entangling and disentangling power of unitary transformations.arXiv preprint arXiv:2505.06774, 2025

    Ammar Daskin. Quantum rnns and lstms through entangling and disentangling power of unitary transformations.arXiv preprint arXiv:2505.06774, 2025

    2025

  94. [102]

    Quantum circuits for partial differential equations in fourier space.arXiv preprint arXiv:2505.16895, 2025

    Michael Lubasch, Yuta Kikuchi, Lewis Wright, and Conor Mc Keever. Quantum circuits for partial differential equations in fourier space.arXiv preprint arXiv:2505.16895, 2025

    2025

  95. [103]

    Tudataset: A collection of benchmark datasets for learning with graphs.arXiv preprint arXiv:2007.08663, 2020

    Christopher Morris, Nils M Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neu- mann. Tudataset: A collection of benchmark datasets for learning with graphs.arXiv preprint arXiv:2007.08663, 2020

    2007 arXiv

  96. [104]

    Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neu- mann

    Christopher Morris, Nils M. Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neu- mann. Tudataset: A collection of benchmark datasets for learning with graphs. InICML 2020 Workshop on Graph Representation Learning and Beyond (GRL+ 2020), 2020

    2020

  97. [105]

    Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019

    2019

  98. [106]

    A fair comparison of graph neural networks for graph classification.arXiv preprint arXiv:1912.09893, 2019

    Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A fair comparison of graph neural networks for graph classification.arXiv preprint arXiv:1912.09893, 2019

    1912 arXiv

  99. [107]

    Saliency-aware regularized graph neural network.Artificial Intelligence, 328:104078, 2024

    Wenjie Pei, Weina Xu, Zongze Wu, Weichao Li, Jinfan Wang, Guangming Lu, and Xiangrong Wang. Saliency-aware regularized graph neural network.Artificial Intelligence, 328:104078, 2024

    2024

  100. [108]

    Geometry-aware edge pooling for graph neural networks.arXiv preprint arXiv:2506.11700, 2025

    Katharina Limbeck, Lydia Mezrag, Guy Wolf, and Bastian Rieck. Geometry-aware edge pooling for graph neural networks.arXiv preprint arXiv:2506.11700, 2025

    2025

  101. [109]

    Identity-aware graph neural networks

    Jiaxuan You, Jonathan M Gomes-Selman, Rex Ying, and Jure Leskovec. Identity-aware graph neural networks. InProceedings of the AAAI conference on artificial intelligence, volume 35, pages 10737– 10745, 2021

    2021

  102. [110]

    An end-to-end deep learning ar- chitecture for graph classification

    Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning ar- chitecture for graph classification. InProceedings of the AAAI conference on artificial intelligence, volume 32, 2018

    2018

  103. [111]

    How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018

    Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018

    2018 arXiv

  104. [112]

    Hi- erarchical graph representation learning with differentiable pooling.Advances in neural information processing systems, 31, 2018

    Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hi- erarchical graph representation learning with differentiable pooling.Advances in neural information processing systems, 31, 2018

    2018

  105. [113]

    Dynamic edge-conditioned filters in convolutional neural networks on graphs

    Martin Simonovsky and Nikos Komodakis. Dynamic edge-conditioned filters in convolutional neural networks on graphs. InProceedings of the IEEE conference on computer vision and pattern recognition, pages 3693–3702, 2017

    2017

  106. [114]

    Entangling and disentangling power of unitary transformations are not equal.Physical review letters, 103(3):030501, 2009

    Noah Linden, John A Smolin, and Andreas Winter. Entangling and disentangling power of unitary transformations are not equal.Physical review letters, 103(3):030501, 2009. 18

    2009

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